Complementarity theory

A complementarity problem is a type of <a href="/facts/Mathematical_optimization/oRn8Iv5I">mathematical optimization</a> problem. It is the problem of optimizing (minimizing or maximizing) a function of two <a href="/facts/Vector_space/M1pxMLx2">vector</a> variables subject to certain requirements (constraints) which include: that the <a href="/facts/Inner_product/HoyjElEy">inner product</a> of the two vectors must equal zero, i.e. they are orthogonal. In particular for finite-dimensional real vector spaces this means that, if one has vectors X and Y with all nonnegative components (xi ≥ 0 and yi ≥ 0 for all 
 
 
 
 i
 
 
 {\displaystyle i}
 
: in the <a href="/facts/First_quadrant/THvuEvBp">first quadrant</a> if 2-dimensional, in the first <a href="/facts/Octant_(solid_geometry)/r1j1h9mz">octant</a> if 3-dimensional), then for each pair of components xi and yi one of the pair must be zero, hence the name complementarity. e.g. X = (1, 0) and Y = (0, 2) are complementary, but X = (1, 1) and Y = (2, 0) are not. A complementarity problem is a special case of a <a href="/facts/Variational_inequality/p3Hy60WS">variational inequality</a>.

Complementarity theory open-in-new

Complementarity theory